Abstract

The research work discussed in this thesis investigated the application of combinatorics and graph theory in the analysis of the partition function of the Ising Model.
Chapter 1 gives a general introduction to the partition function of the Ising Model and the Feynman Identity in the language of graph theory.
Chapter 2 describes and proves combinatorially the Feynman Identity in the special case when there is only one vertex and multiple loops.
Chapter 3 digresses into the number of cycles in a directed graph, along with its application in the special case to derive the analytical expression of the number of non-periodic cycles with positive and negative signs.
Chapter 4 comes back to the general case of the Feynman Identity. The Feynman Identity is applied to several special cases of the graph and a combinatorial identity is established for each case.
Chapter 5 concludes the thesis by summarizing the main ideas in each chapter.